Consider the statement: "For an integer n, if $$n^3 - 1$$ is even, then $$n$$ is odd". The contrapositive statement of this statement is:

We begin with the original conditional statement:

“For an integer $$n$$, if $$n^{3}-1$$ is even, then $$n$$ is odd.”

To form the contrapositive, we first identify its two parts clearly.

Let

$$P : n^{3}-1 \text{ is even}$$

$$Q : n \text{ is odd}$$

The original statement in symbolic form is $$P \Rightarrow Q$$, which reads “If $$P$$, then $$Q$$.”

Now, we recall the definition of a contrapositive. For any implication $$P \Rightarrow Q$$, the contrapositive is obtained by negating both parts and reversing the direction of implication. In symbols, the contrapositive is:

$$\neg Q \Rightarrow \neg P$$

We now write down each negation explicitly.

Negation of $$Q$$:

$$\neg Q : n \text{ is not odd}$$

But for integers, “not odd” is the same as “even.” So we can re-express it as

$$\neg Q : n \text{ is even}$$

Negation of $$P$$:

$$\neg P : n^{3}-1 \text{ is not even}$$

Again, for integers, “not even” means “odd.” Hence we rewrite

$$\neg P : n^{3}-1 \text{ is odd}$$

Substituting these negations into the symbolic form $$\neg Q \Rightarrow \neg P$$, we obtain the contrapositive sentence:

“For an integer $$n$$, if $$n$$ is even, then $$n^{3}-1$$ is odd.”

Looking at the options provided, this matches Option A.

Hence, the correct answer is Option A.